Just as we can define a topological vector space, we can define a "topological X" for any algebraic structure X: we demand that X also be a topological space (usually a Hausdorff space, for convenience), and that the algebraic operations on X be compatible with the topology on X by being continuous.

For a field, we'll demand that addition, negation, multiplication and inversion all be continuous on their domains. For instance, **R** and **Q** are "topological fields", since all arithmetic operations are continuous.

This is actually a useful characterization. But not all topological properties on a topological field turn out to be useful. An example which makes a nice exercise is compactness:

Let F be a topological field which is a compact space. Then F is *finite*.

On most

structures (e.g.

groups), compactness

*extends* finiteness: many things which are true for finite objects also hold for compact objects. The above says we cannot hope for this to hold for fields: the only compact objects

*are* the finite ones!

Interestingly, a proof using non-standard analysis is easier and more intuitive than a "standard" proof. But (unlike the case of Sierpinski's theorem, where the non-standard proof is *much* easier) there is a fairly direct translation of the proof to standard analysis.